If 6 is added to twice a number and this sum is multipled by 2, the result is the same as if the number is multiplied by 3 and 8 is added to the product. what is the number ?

Hey there!

Let's first define this "number" as x. Now, we're going to have to define our keywords. When we say product, that means multiplication, and when it says "the same as", that's signifying equations.

Let's start with the left side.

We have 6 added to twice a number:

2n + 6

Sum multiplied by two:

2(2n+6)

Same as if the number is multiplied by 3:

2(2n+6) = 3n

8 is added to product:

2(2n+6) = 3n + 8

Now, we can use the distributive property, which states that a(b+c) = ab+ac to solve the left side:

2(2n) + 2(6) = 3n + 8
4n + 12 = 3n = 8

Subtract 3n from both sides:

n + 12 = 8

Subtract 12 from both sides:

n = -4

Now, let's plug in -4 to see if it works.

2(2(-4) + 6) = -4(3) + 8
2(-8 + 6) = -12 + 8
2(-2) = -4
-4 = -4

Thus, we can safely say we're right.

Hope this helps!